Contact form in PHP: Allowing the user to send me an email

On my website I have a contact form, where the user can report bugs or submit feature requests. The form contains a subject and message.

I want the form to email me directly, either with PHPMailer or with PHP's built-in mail function.

The issue here is when generating the email, the sender has to be an address I own, not the actual user's email address. The reason for this is obvious, since we shouldn't be allowed to impersonate other people. However I want to be able to email my users back directly from my inbox, which I can't do since I can't use their address to email myself in the script.

What is the best way to construct the contact form then?
The only workaround I can think of is to insert the message + sender address into a database, and then read it manually from there....an epic hassle.